Question

The horizontal asymptotes of the curve $y=\frac{e^x-e^{-x}}{e^x+e^{-x}}$ are

• A. $y=\pm 1$
• B. $y=\pm e$
• C. $y=\pm \frac{1}{e}$
• D. $y=\pm 2e$

Answer 1

The correct option is A $y=\pm 1$

$\underset{x\to \infty }{lim}\frac{e^x-e^{-x}}{e^x+e^{-x}}=\underset{x\to \infty }{lim}\frac{1-e^{-2x}}{1+e^{-2x}}=1$
$\therefore y=1$ is a horizontal asymptote.
$\underset{x\to -\infty }{lim}\frac{e^x-e^{-x}}{e^x+e^{-x}}=\underset{x\to -\infty }{lim}\frac{e^{2x}-1}{e^{2x}+1}=-1$
$\therefore y=-1$ is another horizontal asymptote.